The 78.0 kg climber is supported in the "chimney" by the friction forces exerted on his shoes and back. The static coefficients of friction between his shoes and the wall, and between his back and the wall, are 0.840 and 0.630, respectively. What is the minimum normal force he must exert? Assume the walls are vertical and that friction forces are both at a maximum. Ignore his grip on the rope.

78*g*.63=F1
78*g*.84=F2

F1+F2=78*g

F2-F1=78*g*(.84-.63)

F2=78*g-F1

78*g-2*F2=78*g*(.84-.63)
F2=78*g*(1-.84+.63)/2

F2= 302 N
F1= 463 N

I thought the answer was 302N but the homework website is telling me that I am wrong. Where did I mess up?

Well, it seems like you've made a mistake in your calculations. Let's go through the solution again:

The normal force (N) exerted by the climber can be calculated using the equation:

N = m * g

where m is the mass of the climber and g is the acceleration due to gravity.

Here, m = 78.0 kg, and g is usually taken to be approximately 9.8 m/s^2.

So, N = 78.0 kg * 9.8 m/s^2 = 764.4 N.

Now let's calculate the friction forces:

F1 = μ1 * N
F2 = μ2 * N

where μ1 and μ2 are the static coefficients of friction between the shoes and the wall and between the back and the wall, respectively.

Given that μ1 = 0.840 and μ2 = 0.630, we can substitute these values into the equations:

F1 = 0.840 * 764.4 N = 642.34 N
F2 = 0.630 * 764.4 N = 481.81 N

Now we can check if the sum of the friction forces equals the weight of the climber:

F1 + F2 = 642.34 N + 481.81 N = 1124.15 N

As you can see, the sum of the friction forces is greater than the weight of the climber, which means he is being supported by more than the minimum required normal force.

Therefore, the correct answer would be 1124.15 N, not 302 N.

To determine the minimum normal force the climber must exert, we can find the maximum friction force exerted on his shoes and back.

Given:
Climber's mass, m = 78.0 kg
Coefficient of friction between shoes and wall, μ1 = 0.840
Coefficient of friction between back and wall, μ2 = 0.630
Acceleration due to gravity, g = 9.8 m/s²

To find the maximum friction force on the shoes, we can use the equation:

F1 = μ1 * N1

Similarly, for the back, we have:

F2 = μ2 * N2

Where N1 and N2 are the normal forces exerted on the shoes and back, respectively.

To solve for N1 and N2, we can express the sum of the vertical forces for the climber:

N1 + N2 = m * g

And the difference in the vertical forces:

N2 - N1 = 0

We can solve these two equations to find N1:

2N1 = m * g

N1 = (m * g) / 2

Using the given values, we can calculate N1:

N1 = (78.0 kg * 9.8 m/s²) / 2
N1 = 382.8 N

Since the normal force on the back is equal in magnitude and opposite in direction (Newton's third law), we have:

N2 = N1 = 382.8 N

Therefore, the minimum normal force the climber must exert is 382.8 N, not 302 N as previously calculated.

Apologies for the confusion caused by the previous incorrect solution.

Based on your calculations, it seems that you made an error in one of your equations. Let's go through the steps and identify the mistake.

First, let's define the variables:
F1 = Normal force exerted on the climber's back
F2 = Normal force exerted on the climber's shoes
g = acceleration due to gravity (approximately 9.8 m/s^2)

Now, we can set up the equations:

1. Friction force on the climber's back:
F1 = 78 kg * g * 0.63

2. Friction force on the climber's shoes:
F2 = 78 kg * g * 0.84

3. Total upward force on the climber:
F_total_upward = F1 + F2

According to your calculations, you correctly determined the values of F1 and F2 as follows:

F1 = 78 kg * g * 0.63 = 463 N
F2 = 78 kg * g * 0.84 = 611 N

Now, let's calculate the correct value for the total upward force exerted by the climber:

F_total_upward = F1 + F2 = 463 N + 611 N = 1074 N

Therefore, the correct minimum normal force the climber must exert is 1074 N, not 302 N.

Please double-check your calculations and ensure that you've correctly calculated the friction forces and added them together to find the total upward force.